Armature with automatic weights problem. Object not following the bone properly
When Blender automatically generated the weight for the bones it must have not completely covered the area of the bone.  What you need to do is select the mesh and go into weight paint mode.  Select the vertex group for the bone that is having problems, set the weight to 1.000, and paint the area around the bone.  That should fix your problem.  Painting may be needed on more bones than just one.